Physics AS unit 2, section 7.1 - vectors and scalars

In this section:

• what is a vector quantity?
• what is a scalar quantity?
• what ia a displacement vector?
• what's the difference between velocity and speed?
• is acceleration a vector or scalar quantity?
• how can I add vectors together to get a resultant vector?
• how can I use Pythagoras' theorem to add perpendicular vectors ?
• how do I find the magnitude and direction of a resultant vector?
• how can I break a vector down into perpendicular components?

Vectors and scalars

A vector quantity has a size and a direction. A scalar quantity has only size without direction.
Examples:

. . . vector quantities . . .	. . . scalar quantities. . .
displacement	distance
weight	mass
velocity	speed
acceleration	density
force

Representation of vectors

Vectors can be represented by arrows, usually drawn in the plane (2D). The length of the arrow represents the size of the vector quantity and the direction of the vector is represented by the direction of the arrow. Here are some examples drawn on 5mm squared exercise paper:

Explore with this mini vector plotting widget:

Vector notation

You may recall from GCSE mathematics that vectors are represented in print either by a bold-print lower-case letter such as v or the upper-case (capital) letters which refer to the points at the beginning and end of the vector with an arrow above:
$\displaystyle \oddsidemargin \vec{PQ}$
In the AQA physics text, they use bold-print upper-case letters denoting the beginning and end of the vector such as in PQ without the arrow. This is not good style in a maths exam.
In handwritten notes the lower-case vector v is written v

Addition of vectors

Adding vectors using a scale diagram is easy: simply draw the two vectors end-to-end. Start the second vector where the first finishes.
The answer to your addition is called the resultant.

solution:

Explore the addition of vectors with the widget:

Perpendicular vectors and Pythagoras' theorem:

Where the two vectors are either parallel (in the same direction or opposite directions) or perpendicular (at right angles) you should be able to easily calculate the resultant without having to produce an accurate scale diagram. However, even here it's probably worth drawing a sketch to be sure you've got all the vectors pointing the right way.
When the vectors are parallel just add or subtract their sizes (magnitudes).
When the vectors are perpendicular you will be able to use Pythagoras' theorem - I bet your maths teacher told you it would be useful one day!
In this widget, we force the vectors to be perpendicular. See if you can see what happens to the magnitudes:

Suppose, as is the default for the widget, we have perpendicular forces of 3N and 4N. Their resultant magnitude is:
$\displaystyle R = \sqrt {F_1^2 + F_2^2} = \sqrt {3^2 + 4^2} = \sqrt {9 + 16} = \sqrt {25} = 5$

You can find the angle of the resultant using trigonometry.
You may remember that:
$\displaystyle \tan {\theta} = \frac {opposite} {adjacent}$
so here we use:
$\displaystyle \tan {\theta} = \frac {4} {3}$
$\displaystyle \theta = \arctan {\frac {4} {3}}$
$\displaystyle \theta = 36.87 \circ$
And in general given two perpendicular forces F1 and F2 the angle J between the resultant and F1 is given by:
$\displaystyle \tan {\theta} = \frac {F_2} {F_1}$
This widget allows you to check that you have calculated these correctly, but you need to develop the mathematical skills to find resultant magnitudes and directions using Pythagoras' theorem and tan-1

Resolving vectors using trigonometry

In a similar way to adding vectors above, you can resolve a vector where you know its magnitude and direction into horizonatal and vertical components using trigonometry.

Homework

The homework task is to complete the following questions by Monday 19 September 2011:

Homework hints

1. Look at the list of vector and scalar quantities above. Magnetic field is the tricky one. Google "is magnetic field a vector?"
2. Find magnitude of vectors by using Pythagoras' theorem. This still works in 3D for questions (b) and (d) but you'll get three terms inside the root
3. Vector AB is found by subtracting the coordinates of A from the coordinates of B. This also works in 3D. See the picture below.
4. Draw it first. Find OX = ½OA and OY = ½OB. Now find AB = OB - OA and XY = OY - OX. Compare. If one is a multiple ot the other they're parallel.
5. Mis-print on handout. See the version in the Word document above. Now apply the hint for question 4.
6. The opposite sides in a parallelogram are equal in length and in the same direction. Therefore the vectors along opposite sides are equal.
7. Try the Geogebra applet at http://www.geogebratube.org/student/m465. You may need to download some freeware from Geogebra. Let me know if this works, please.
8. If AG = 2GM then AG is one third of AM. Draw it. Add points on the other sides. Call them something sensible like N and P. You should find the same thing happens for them and the point one third along these gives the same position vector as OG. This is called the centroid of the triangle You can play with a Geogebra applet at http://www.geogebratube.org/student/m466. I can see there have been 21 views of the other one already! Great stuff.
9. Try taking two routes to Z: OA + kAB and OX + mXY. Equate k and m using coefficients of a and b. Get simultaneous equations. Solve for k, m.
10. X,Y, Z lie on a straight line implies XY = kYZ for some value of k.
11. ABCDEF is a regular hexagon. Think about what this means about the lengths and angles.

More hints:
The full solution set:

Homework 2:

In the second homework I asked you to solve the following problems:

homework 2 solutions: